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Quantum codes of minimum distance two

Rains, Eric M. (1999) Quantum codes of minimum distance two. IEEE Transactions on Information Theory, 45 (1). pp. 266-271. ISSN 0018-9448. doi:10.1109/18.746807.

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It is reasonable to expect the theory of quantum codes to be simplified in the case of codes of minimum distance 2; thus it makes sense to examine such codes in the hopes that techniques that prove effective there will generalize. With this in mind, we present a number of results on codes of minimum distance 2. We first compute the linear programming bound on the dimension of such a code, then show that this bound can only be attained when the code either is of even length, or is of length 3 or 5. We next consider questions of uniqueness, showing that the optimal code of length 2 or 1 is unique (implying that the well-known one-qubit-in-five single-error correcting code is unique), and presenting nonadditive optimal codes of all greater even lengths. Finally, we compute the full automorphism group of the more important distance 2 codes, allowing us to determine the full automorphism group of any GF(4)-linear code.

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Additional Information:© 1999 IEEE. Manuscript received May 26, 1997; revised March 4, 1998. The author would like to thank A. R. Calderbank, P. Shor, and N. Sloane for many helpful conversations, as well as the anonymous referees for helpful comments.
Subject Keywords:Automorphisms, classification, quantum codes, uniqueness
Issue or Number:1
Record Number:CaltechAUTHORS:20170926-144218757
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Official Citation:E. M. Rains, "Quantum codes of minimum distance two," in IEEE Transactions on Information Theory, vol. 45, no. 1, pp. 266-271, Jan 1999. doi: 10.1109/18.746807 URL:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:81848
Deposited By: Tony Diaz
Deposited On:26 Sep 2017 22:43
Last Modified:15 Nov 2021 19:46

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